The pages of a transverse disk open book whose binding has Conley–Zehnder index $\ge3$ are disk-like global surfaces of section

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HryniewiczCriterion.isDiskLikeGlobalSectionMap_of_transverse_open_book

by Mazecto · Oct 20, 2026 · Mathlib 0df444a (Lean v4.33.1)

dynamical-systemsgeometry-topologysymplectic-geometry

Let $S=H^{-1}(1)\subset\mathbb{R}^4$ be a strictly star-shaped level and let $P=(x,T)$ be a periodic orbit on $S$ with $\mu_{CZ}(P)\ge3$ . Let $(\mathrm{page}_\theta)_{\theta\in\mathbb{R}}$ be a transverse disk open book on $S$ with binding $P$ . Then every page $\mathrm{page}_\theta(\bar{\mathbb D})$ is a disk-like global surface of section bounded by $P$ (IsDiskLikeGlobalSectionMap): every trajectory of $X_H$ on $S$ not contained in $x(\mathbb{R})$ meets $\mathrm{page}_\theta(\bar{\mathbb D})$ at arbitrarily large positive and arbitrarily large negative times.

A transverse disk open book on $S=H^{-1}(1)$ with binding $P=(x,T)$ is a family of maps $\mathrm{page}_\theta:\mathbb{R}^2\to\mathbb{R}^4$ , $\theta\in\mathbb{R}$ , such that

$(\theta,u)\mapsto\mathrm{page}_\theta(u)$ is $C^\infty$ and $1$ -periodic in $\theta$ ;
each $\mathrm{page}_\theta$ restricts to a smooth embedding of the closed unit disk $\bar{\mathbb D}$ into $S$ (IsSmoothDiskEmbedding, injective differential up to the boundary);
the boundary circle runs along $P$ : $\mathrm{page}_\theta(\cos2\pi t,\sin2\pi t)=x(Tt)$ ;
$X_H$ is transverse to every page in the interior: $X_H(\mathrm{page}_\theta(u))\notin\operatorname{im}D\mathrm{page}_\theta(u)$ for $|u|<1$ ;
$(\theta,u)\mapsto\mathrm{page}_\theta(u)$ maps $[0,1)\times\mathbb D$ bijectively onto $S\setminus x(\mathbb{R})$ , with injective differential at every $(\theta,u)$ , $|u|<1$ .

This is an adapted open book decomposition (AdaptedDiskOpenBook) without the requirement that the pages be global surfaces of section and without the orientation condition on the pages.

$\mu_{CZ}(P)\ge3$ means 3 ≤ czIndexOfPath (linearizedXiPath H P Y) for every linearized flow $Y$ of $X_H$ along $x$ (the linearized flow on $\xi$ in the global frame $(Z_1,Z_2)$ , with the lower semicontinuous extension of the index to degenerate orbits). No non-degeneracy is assumed. The statement turns a transverse disk open book whose binding has index at least $3$ into a family of disk-like global surfaces of section with common boundary $P$ .

Preamble

import Definitions.Def_HryniewiczCriterion_ConleyZehnder
import Definitions.Def_HryniewiczCriterion_GlobalSection
import Definitions.Def_HryniewiczCriterion_SelfLinking

open HryniewiczCriterion
open scoped ContDiff

Formal statement

theorem HryniewiczCriterion.isDiskLikeGlobalSectionMap_of_transverse_open_book (H : R4 → ℝ) (hS : IsStrictlyStarShapedLevel H) (P : PeriodicOrbit H)
    (hcz : ∀ Y : ℝ → (R4 →L[ℝ] R4), IsLinearizedFlow H P.x Y →
      3 ≤ czIndexOfPath (linearizedXiPath H P Y))
    (page : ℝ → Plane → R4)
    (hob : ContDiff ℝ ∞ (fun z : ℝ × Plane => page z.1 z.2) ∧
      (∀ θ u, page (θ + 1) u = page θ u) ∧
      (∀ θ, IsSmoothDiskEmbedding (page θ)) ∧
      (∀ θ, page θ '' closedUnitDisk ⊆ energySurface H) ∧
      (∀ θ t, page θ (circlePoint t) = P.x (P.T * t)) ∧
      (∀ θ, ∀ u ∈ openUnitDisk,
        hamiltonianVectorField H (page θ u) ∉ Set.range (fderiv ℝ (page θ) u)) ∧
      (∀ y ∈ energySurface H, y ∉ P.image →
        ∃! z : ℝ × Plane, z.1 ∈ Set.Ico (0 : ℝ) 1 ∧ z.2 ∈ openUnitDisk ∧ page z.1 z.2 = y) ∧
      (∀ θ : ℝ, ∀ u ∈ openUnitDisk,
        Function.Injective (fderiv ℝ (fun z : ℝ × Plane => page z.1 z.2) (θ, u)))) :
    ∀ θ : ℝ, IsDiskLikeGlobalSectionMap H P (page θ) := by sorry

Source

U. Hryniewicz, Systems of global surfaces of section for dynamically convex Reeb flows on the 3-sphere, J. Symplectic Geom. 12 (2014) 791-862, arXiv:1105.2077, Theorem 3.11 (p. 25); H. Hofer, K. Wysocki, E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. 148 (1998) 197-289, Theorem 2.5.

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The pages of a transverse disk open book whose binding has Conley–Zehnder index ≥3\ge3≥3 are disk-like global surfaces of section

The pages of a transverse disk open book whose binding has Conley–Zehnder index $\ge3$ are disk-like global surfaces of section